Skip to main content

Full text of "Treatise On Analysis Vol-Ii"

See other formats


3    SPECTRUM OF A COMMUTATIVE BANACH ALGEBRA       319

x(A) = C, the quotient algebra A/rn is a field isomorphic to C, hence m is
maximal. Conversely, let n be a maximal ideal of A. The fact that n is closed is
a consequence of the following lemma:

(15.3.1.1)    If a 7^ A is an ideal in A, then its closure a is an ideal ^A.

For the complement (Ja contains the set G of invertible elements of A,
which is open in A (15.2.4(i)), and hence G is contained in the complement
of a.

Applied to the maximal ideal n, this lemma shows that n = n, because
n r> n and n ^ A. The quotient normed algebra (15.1) A/n is therefore a
Banach algebra (12.14.9) which is a field because n is maximal, hence bv the
Gelfand-Mazur theorem (15.2.5) is isomorphic to C. In other words, there
exists a unique homomorphism % : A ~> C such that %(e) = 1 and x"1^) = n,
and the proof of (iii) is complete.

(15.3.2) Let A be a Banach algebra with unit element e ^ 0. The set X(A) of
characters of A is a subset of the unit ball \\x'\\ <; 1 in the dual A' of the Banach
space A, and is closed in A with respect to the weak topology (12.15). If A is
separable then X(A) is metrizable and compact for the weak topology.

In view of (12,15.9) we need only prove that X(A) is a closed subset of A',
or equivalently that if u lies in the closure of X(A) in A' for the weak topology,
then <xy, u) = <jc, w><j, w> for all x, y e A, and <e, M> = 1. But this follows
from the continuity of the mapping I/H* <jc, w> of A' into C (for each x e A)
and the principle of extension of identities.

The set X(A), endowed with the topology induced by the weak topology
on A, is called the spectrum of A. For each x e A, the mapping %*-+x(x) of
X(A) into C is denoted by &x or ^Ax, and is called the Gelfand transform of
x\ the mapping x^>^x of A into CX(A) is called the Gelfand transformation.
Hence we have by definition, for all x e A and all x 6 X(A),

(15.3.3)                                   (*)(*) = *(*)

(15.3.4)    Let A be a separable commutative Banach algebra with unit element
e^O.

(i) The Gelfand transformation x\-*9x is a continuous homomorphism of
the Banach algebra A into the Banach algebra ^C(X(A)) such that \\9x\\ =
p(x) ^ ||x|| and such that %e is the constant function 1.

(ii) The set of values of the continuous function <&x on X(A) is equal to
SpA(x). In particular, for x to be invertible it is necessary and sufficient that
<&x should not vanish on X(A). in C. Show that